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Nonparametric inference of doubly stochastic Poisson process data via the kernel method. (English) Zbl 1220.62037

Summary: Doubly stochastic Poisson processes, also known as the Cox processes, frequently occur in various scientific fields. In this article, motivated primarily by analyzing Cox process data in biophysics, we propose a nonparametric kernel-based inference method. We conduct a detailed study, including an asymptotic analysis, of the proposed method, and provide guidelines for its practical use, introducing a fast and stable regression method for bandwidth selection. We apply our method to real photon arrival data from recent single-molecule biophysical experiments, investigating proteins’ conformational dynamics. Our result shows that conformational fluctuation is widely present in protein systems, and that the fluctuation covers a broad range of time scales, highlighting the dynamic and complex nature of proteins structure.

MSC:

62G07 Density estimation
62M09 Non-Markovian processes: estimation
92C05 Biophysics
62E20 Asymptotic distribution theory in statistics
62P35 Applications of statistics to physics

Software:

KernSmooth; sm
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References:

[1] Barbieri, R., Wilson, M. A., Frank, L. M. and Brown, E. N. (2005). An analysis of hippocampal spatio-temporal representations using a Bayesian algorithm for neural spike train decoding. IEEE Transactions on Neural Systems and Rehabilitation Engineering 13 131-136.
[2] Berman, M. and Diggle, P. J. (1989). Estimating weighted integrals of the second-order intensity of a spatial point process. J. Roy. Statit. Soc. Ser. B 51 81-92. · Zbl 0671.62043
[3] Bialek, W., Rieke, F., Steveninck, R. and Warland, D. (1991). Reading a neural code. Science 252 1854-1857.
[4] Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley, New York. · Zbl 0944.60003
[5] Bowman, A. and Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis: The Kernel Approach With S-PLUS Illustrations . Oxford Univ. Press, New York. · Zbl 0889.62027
[6] Carroll, B. and Ostlie, D. (2007). An Introduction to Modern Astrophysics , 2nd ed. Benjamin Cummings, Reading, MA.
[7] Cox, D. R. (1955a). The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Proc. Camb. Phil. Soc. 51 433-441. · Zbl 0067.10902
[8] Cox, D. R. (1955b). Some statistical methods connected with series of events. J. R. Statist. Soc. Ser. B 17 129-164. · Zbl 0067.37403
[9] Cox, D. R. and Isham, V. (1980). Point Processes . Chapman and Hall, London. · Zbl 0441.60053
[10] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes . Springer, New York. · Zbl 0657.60069
[11] Diggle, P. (1985). A kernel method for smoothing point process data. Appl. Statist. 34 138-147. · Zbl 0584.62140
[12] Diggle, P. (2003). Statistical Analysis of Spatial Point Patterns . Oxford Univ. Press, New York. · Zbl 1021.62076
[13] English, B., Min, W., van Oijen, A. M., Lee, K. T., Luo, G., Sun, H., Cherayil, B. J., Kou, S. C. and Xie, X. S. (2006). Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited. Nature Chem. Biol. 2 87-94.
[14] Eubank, R. L. (1988). Nonparametric Regression and Spline Smoothing . Marcel Dekker, New York. · Zbl 0702.62036
[15] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications . Chapman and Hall, London. · Zbl 0873.62037
[16] Gerstner, W. and Kistler, W. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity . Cambridge Univ. Press. · Zbl 1100.92501
[17] Grund, B., Hall, P. and Marron, J. S. (1994). Loss and risk in smoothing parameter selection. Nonparametr. Stat. 4 133-147. · Zbl 1380.62148
[18] Guan, Y. (2007). A composite likelihood cross-validation approach in selecting bandwidth for the estimation of the pair Correlation function. Scand. Statist. 34 336-346. · Zbl 1142.62072
[19] Guan, Y., Sherman, M. and Calvin, J. A. (2004). A nonparametric test for spatial isotropy using subsampling. J. Amer. Statist. Assoc. 99 810-821. · Zbl 1117.62348
[20] Guan, Y., Sherman, M. and Calvin, J. A. (2006). Assessing isotropy for spatial point processes. Biometrics 62 126-134. · Zbl 1091.62096
[21] Härdle, W. (1990). Applied Nonparametric Regression . Cambridge Univ. Press. · Zbl 0714.62030
[22] Hawkes, A. G. (2005). Ion channel modeling. In Encyclopedia of Biostatistics , 2nd ed. (P. Armitage and T. Colton, eds.) 4 2625-2632. Wiley, Chichester.
[23] Ibragimov, I. A. and Rozanov, Y. A. (1978). Gaussian Random Processes . New York, Springer. · Zbl 0392.60037
[24] Jones, M. C., Marron, J. S. and Sheather, S. J. (1996). A brief survey of bandwidth selection for density estimation. J. Amer. Statist. Assoc. 91 401-407. · Zbl 0873.62040
[25] Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes . Academic Press, New York. · Zbl 0469.60001
[26] Karr, A. F. (1991). Point Processes and Their Statistical Inference , 2nd ed. Marcel Dekker, New York. · Zbl 0733.62088
[27] Kou, S. C. (2008a). Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins. Ann. Appl. Statist. 2 501-535. · Zbl 1400.62272
[28] Kou, S. C. (2008b). Stochastic networks in nanoscale biophysics: Modeling enzymatic reaction of a single protein. J. Amer. Statist. Assoc. 103 961-975. · Zbl 1205.62172
[29] Kou, S. C. (2009). A selective view of stochastic inference and modeling problems in nanoscale biophysics. Science in China A 52 1181-1211. · Zbl 1211.62185
[30] Kou, S. C., Cherayil, B., Min, W., English, B. and Xie, X. S. (2005b). Single-molecule Michaelis-Menten equations. J. Phys. Chem. B 109 19068-19081.
[31] Kou, S. C. and Xie, X. S. (2004). Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule. Phys. Rev. Lett. 93 180603(1)-180603(4).
[32] Kou, S. C., Xie, X. S. and Liu, J. S. (2005a). Bayesian analysis of single-molecule experimental data (with discussion) J. Roy. Statist. Soc. Ser. C 54 469-506. · Zbl 05188696
[33] Krichevsky, O. and Bonnet, G. (2002). Fluorescence correlation spectroscopy: The technique and its applications. Report on Progress in Physics 65 251-297.
[34] Lerch, H., Rigler, R. and Mikhailov, A. (2005). Functional conformational motions in the turnover cycle of cholesterol oxidase. Proc. Natl. Acad. Sci. USA 102 10807-10812.
[35] Lu, H. P., Xun, L. and Xie, X. S. (1998). Single-molecule enzymatic dynamics. Science 282 1877-1882.
[36] Marron, J. S. and Tsybakov, A. B. (1995). Visual error criteria for qualitative smoothing. J. Amer. Statist. Assoc. 90 499-507. · Zbl 0826.62026
[37] Marron, J. S. and Wand, M. P. (1992). Exact mean integrated squared error. Ann. Statist. 20 712-736. · Zbl 0746.62040
[38] Meegan, C. A., Fishman, G. J., Wilson, R. B., Paciesas, W. S., Pendleton, G. N., Horack, J. M., Brock, M. N. and Kouveliotou, C. (1992). Spatial distribution of gamma ray bursts observed by BATSE. Nature 355 142-145.
[39] Min, W., English, B., Luo, G., Cherayil, B., Kou, S. C. and Xie, X. S. (2005a). Fluctuating enzymes: Lessons from single-molecule studies. Accounts of Chemical Research 38 923-931.
[40] Min, W., Luo, G., Cherayil, B., Kou, S. C. and Xie, X. S. (2005b). Observation of a power law memory kernel for fluctuations within a single protein molecule. Phys. Rev. Lett. 94 198302(1)-198302(4).
[41] Moller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes . Chapman and Hall, New York.
[42] Mukamel, S. (1995). Principle of Nonlinear Optical Spectroscopy . Oxford Univ. Press.
[43] Müller, H. G. (1988). Nonparametric Regression Analysis of Longitudinal Data . Springer, New York. · Zbl 0664.62031
[44] Park, B. U. and Turlach, B. A. (1992). Practical performances of several data-driven bandwidth selectors (with discussion). Comput. Statist. 7 251-285. · Zbl 0775.62100
[45] Parzen, E. (1962). Stochastic Processes . Holden Day, Inc., San Francisco, CA. · Zbl 0107.12301
[46] Reilly, P. D. and Skinner, J. L. (1994). Spectroscopy of a chromophore coupled to a lattice of dynamic two-level systems. J. Chem. Phy s. 101 959-973.
[47] Rieke, F., Warland, D., de Ruyter van Steveninck, R. and Bialek, W. (1996). Spikes: Exploring the Neural Code . MIT Press, Cambridge, MA. · Zbl 0912.92004
[48] Sakmann, B. and Neher, E. (1995). Single Channel Recording , 2nd ed. Plenum Press, New York.
[49] Scargle, J. D. (1998). Studies in astronomical time series analysis. V. Bayesian blocks, a new method to analyze structure in photon counting data. Astrophys. J. 504 405-418.
[50] Schenter, G. K., Lu, H. P. and Xie, X. S. (1999). Statistical analyses and theoretical models of single-molecule enzymatic dynamics. J. Phys. Chem. A 103 10477-10488.
[51] Scott, D. W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization . Wiley, New York. · Zbl 0850.62006
[52] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis . Chapman and Hall, London. · Zbl 0617.62042
[53] Stoyan, D. and Stoyan, H. (1994). Fractals, Random Shapes and Point Fields . Wiley, New York. · Zbl 0828.62085
[54] Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrsch. verw. Gebiete 31 287-302. · Zbl 0303.60033
[55] Wahba, G. (1990). Spline Models for Observational Data . SIAM, Philadelphia. · Zbl 0813.62001
[56] Wand, M. P. and Jones, M. C. (1994). Kernel Smoothing . Chapman and Hall, London. · Zbl 0854.62043
[57] Wang, H., Lin, S., Allen, J. P., Williams, J. C., Blankert, S., Laser C. and Woodbury, N. W. (2007). Protein dynamics control the kinetics of initial electron transfer in photosynthesis. Science 316 747-750.
[58] Weiss, S. (2000). Measuring conformational dynamics of biomolecules by single molecule fluorescence spectroscopy. Nature Struct. Biol. 7 724-729.
[59] Whitt, W. (2002). Stochastic-Process Limits . Springer, New York. · Zbl 0993.60001
[60] Yang, H. and Xie, X. S. (2002a). Statistical approaches for probing single-molecule dynamics photon by photon. Chem. Phys. 284 423-437.
[61] Yang, H. and Xie, X. S. (2002b). Probing single molecule dynamics photon by photon. J. Chem. Phys. 117 10965-10979.
[62] Yang, H., Luo, G., Karnchanaphanurach, P., Louise, T.-M., Rech, I., Cova, S., Xun, L. and Xie, X. S. (2003). Protein conformational dynamics probed by single-molecule electron transfer. Science 302 262-266.
[63] Zhang, T. and Kou, S. C. (2010). Supplement to “Nonparametric inference of doubly stochastic Poisson process data via the kernel method.” DOI: . · Zbl 1220.62037
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